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A265429 Total number of ON (black) cells after n iterations of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.

%I A265429 #35 Feb 16 2025 08:33:27
%S A265429 1,3,5,9,13,18,23,30,37,45,53,63,73,84,95,108,121,135,149,165,181,198,
%T A265429 215,234,253,273,293,315,337,360,383,408,433,459,485,513,541,570,599,
%U A265429 630,661,693,725,759,793,828,863,900,937,975,1013,1053,1093,1134
%N A265429 Total number of ON (black) cells after n iterations of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.
%D A265429 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A265429 Robert Price, <a href="/A265429/b265429.txt">Table of n, a(n) for n = 0..999</a>
%H A265429 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A265429 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A265429 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A265429 Conjectures from _Colin Barker_, Dec 09 2015 and Apr 16 2019: (Start)
%F A265429 a(n) = (1/16)*(6*n^2 + 24*n - 3*(-1)^n + 2*(-i)^n + 2*i^n + 15) where i = sqrt(-1).
%F A265429 G.f.: (1 + x + 2*x^3 - x^4) / ((1-x)^3*(1+x)*(1+x^2)).
%F A265429 (End)
%F A265429 Conjecture: the sequence consists of all numbers k > 0 such that floor(sqrt(8*(k+1)/3)) != floor(sqrt(8*k/3)). - _Gevorg Hmayakyan_, Sep 01 2019
%e A265429 From _Michael De Vlieger_, Dec 09 2015: (Start)
%e A265429 First 12 rows, replacing "0" with ".", ignoring "0" outside of range of 1's for better visibility of ON cells, followed by total number of ON cells per row, and running total up to that row:
%e A265429 1                          =  1 ->   1
%e A265429 1 1                        =  2 ->   3
%e A265429 1 . 1                      =  2 ->   5
%e A265429 1 1 1 1                    =  4 ->   9
%e A265429 1 1 1 . 1                  =  4 ->  13
%e A265429 1 1 . 1 1 1                =  5 ->  18
%e A265429 1 . 1 1 1 . 1              =  5 ->  23
%e A265429 1 1 1 1 . 1 1 1            =  7 ->  30
%e A265429 1 1 1 . 1 1 1 . 1          =  7 ->  37
%e A265429 1 1 . 1 1 1 . 1 1 1        =  8 ->  45
%e A265429 1 . 1 1 1 . 1 1 1 . 1      =  8 ->  53
%e A265429 1 1 1 1 . 1 1 1 . 1 1 1    = 10 ->  63
%e A265429 1 1 1 . 1 1 1 . 1 1 1 . 1  = 10 ->  72
%e A265429 (End)
%t A265429 rule = 188; rows = 30; Table[Total[Take[Table[Total[Table[Take[CellularAutomaton[rule,{{1},0},rows-1,{All,All}][[k]],{rows-k+1,rows+k-1}],{k,1,rows}][[k]]],{k,1,rows}],k]],{k,1,rows}]
%t A265429 Accumulate[Count[#, n_ /; n == 1] & /@ CellularAutomaton[188, {{1}, 0}, 53]] (* _Michael De Vlieger_, Dec 09 2015 *)
%Y A265429 Cf. A118174.
%K A265429 nonn,easy
%O A265429 0,2
%A A265429 _Robert Price_, Dec 08 2015